Ryan Budney and Fred Cohen

Abstract

Consider the space of long knots in ℝn,
Kn,1. This
is the space of knots as studied by V Vassiliev. Based on previous work
[Budney: Topology 46 (2007) 1–27], [Cohen, Lada and May: Springer
Lecture Notes 533 (1976)] it follows that the rational homology of
K3,1 is free
Gerstenhaber–Poisson algebra. A partial description of a basis is given here. In addition,
the mod–p
homology of this space is a free, restricted Gerstenhaber–Poisson algebra.
Recursive application of this theorem allows us to deduce that there is
p–torsion of all orders in
the integral homology of K3,1.

This leads to some natural questions about the homotopy type of the space of long
knots in ℝn
for n>3,
as well as consequences for the space of smooth embeddings of
S1 in
S3 and
embeddings of S1
in ℝ3.